OFFSET
0,3
COMMENTS
For the quaternion Q = 2+j+k, Q^n = r(n) + a(n)*(j+k). The sequence of real-parts r(n) is A266046. - Stanislav Sykora, Dec 20 2015
LINKS
Stanislav Sykora, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (4,-6).
FORMULA
G.f.: x/(1-4*x+6*x^2). - Philippe Deléham, Oct 12 2011
2*a(n)^2 + A266046(n)^2 = 6^n. - Stanislav Sykora, Dec 20 2015
From G. C. Greubel, Jan 10 2024: (Start)
a(n) = 6^((n-1)/2)*ChebyshevU(n-1, sqrt(2/3)).
E.g.f.: (1/sqrt(2))*exp(2*x)*sin(sqrt(2)*x). (End)
a(n) = (i*sqrt(2)/4)*((2 - i*sqrt(2))^n - (2 + i*sqrt(2))^n), where i = sqrt(-1). - Taras Goy, Jan 03 2025
MAPLE
w := I*sqrt(2): a := n -> (w/4)*((2 - w)^n - (2 + w)^n):
seq(simplify(a(n)), n = 0..20); # (after Taras Goy), Peter Luschny, Jan 03 2025
MATHEMATICA
LinearRecurrence[{4, -6}, {0, 1}, 50]
PROG
(PARI) a(n)=([0, 1; 0, 0]*[0, -6; 1, 4]^n)[1, 1] \\ Charles R Greathouse IV, May 31 2011
(Magma) [n le 2 select n-1 else 4*Self(n-1) -6*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 10 2024
(SageMath)
A190965=BinaryRecurrenceSequence(4, -6, 0, 1)
[A190965(n) for n in range(41)] # G. C. Greubel, Jan 10 2024
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Vladimir Joseph Stephan Orlovsky, May 24 2011
STATUS
approved